Currently, it's impossible to gracefully shutdown a .NET Core based App Service in Azure. Whenever the stop button is clicked (or the application unloads itself or detects the app_offline.htm) the process gets killed with an error (visible under the Application Events area): Failed to gracefully shutdown process ''. This is true for the most basic template (created with dotnet new).

The .NET version seems to be up to date, but I'm not 100% sure:

> dotnet --version
1.0.1

> dotnet --info
.NET Command Line Tools (1.0.1)

Product Information:
 Version:            1.0.1
 Commit SHA-1 hash:  005db40cd1

Runtime Environment:
 OS Name:     Windows
 OS Version:  6.2.9200
 OS Platform: Windows
 RID:         win8-x86
 Base Path:   D:\Program Files (x86)\dotnet\sdk\1.0.1